This is an old revision of the document!
Rational numbers
Framework
$\dots,\,-\tfrac{4}{3},\,0,\,\tfrac{1}{17},\,1,\,7.528,\,9001,\dots$
The rational numbers ${\mathbb{Q}}$ can be defined as the field of characteristic 0 which has no proper subfield. In less primitive notions, it's the field of fractions for the integral domain of natural numbers. The second order theory of rationals (see the note below) describes a countable collection.
The rationals can also be set up straight forwardly from tuples of natural numbers.
Discussion
Theorems
For all $m$
| $\sum_{k=0}^m x^k = \dfrac{1}{1-x}(1-x^{m+1})$ |
|---|
| $\sum_{k=0}^m (1-y)^k = \dfrac{1}{y}-\dfrac{1}{y}(1-y)^{m+1}$ |
Logic
In first order logic, being of characteristic zero (“$\forall n.\,(1+1+\dots+1)_{n\ \text{times}}\neq 0$”) requires an axiom schema. But even the induction axiom of the Peano axioms requires a schema.
Also, due to the Löwenheim–Skolem theorem, all theories of infinite structures (e.g. ${\mathbb N}, {\mathbb Q}, {\mathbb R}$) have bad properties in first order logic. For $\mathbb Q$, there is a first-order theory of fields and one can also characterize characteristic 0, however the notion of “proper subfield” is elusive. One needs second-order logical to capture it categorically (=all possible models are isomorphic), see the references.
References
Wikipedia: Field of fractions, Löwenheim–Skolem theorem
Math StackExchange: Axiomatic characterization of the rational numbers, What is the first order axiom characterizing a field having characteristic zero?
Related
